Graded Multiplicities in the Kostant-Rallis Setting

2023/12/18 by Frohmader, Andrew
#Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2312.11295

Abstract

This paper contains two main results. First, we provide combinatorial branching rules for GL_n \downarrow O_n and GL_2n \downarrow Sp_2n extending the Littlewood restriction rules. Second, we use these branching rules and the combinatorics of GL_n-crystals to derive a formula for the graded multiplicity of a K-type in the regular functions on the K-nilpotent cone for GL(n, ℝ), GL(n, ℂ) and GL(n, ℍ). Due to work of Schmid and Vilonen, these graded multiplicities determine the Hodge K-character of the spherical principal series with infinitesimal character 0.

Related

• On Higher Multiplicity upon Restriction from GL(n) to GL(n-1)
• Tensor product varieties and crystals. GL case
• Generalized Littlewood-Richardson coefficients for branching rules of GL(n) and extremal weight crystals
• Graded characters of modules supported in the closure of a nilpotent conjugacy class
• On Multiplicity Formula for Spherical Varieties